JEE Advanced 2020 Full Test 5 Paper 1

Let two perpendicular chords through A(x1,y1) be PQ and RS.
Equation of PQ is ‌

x−x1

cos‌θ

=‌

y−y1

sin‌θ

=r where tan‌θ= slope of PQ
Any point on this line may be taken as (x1+r‌cos‌θ,y1+ r‌sin‌θ). As the point lies on y2=4ax
∴‌‌(y1+r‌sin‌θ)2=4a(x1+r‌cos‌θ)
⇒r2sin2θ+2y1‌sin‌θ⋅r−4a‌cos‌θ‌r+y12−4ax1=0
∴‌‌r1r2=‌

−(y12−4ax1)

sin2θ

Similarly r3r4=‌

−(y12−4ax1)

sin2α

where tan‌α= slope of RS Since PQ is perpendicular to RS,
∴‌‌α=90+θ or θ−90 In either case: sin2α=cos2θ
Now ‌

1

r1r2

+‌

1

r3r4

=‌

sin2θ+cos2θ

−(y12−4ax1)

=‌

1

−(y12−4ax1)

which is constant